I want to get the assembly version of my application that is being built to display on the web site.

You can use Reflector, ILDASM, Jetbrains dotpeek or ILSpy to get the assembly version.

You usually can find ILDASM in C:\Program Files (x86)\Microsoft SDKs\Windows\v8.1A\bin\NETFX 4.5.1 Tools\ildasm.exe (where v8.1A is the version of the Windows SDK installed).

ILDASM: Ildasmtooltoseeversion

how do I set these properties during my build? You need to set extended properties in the form of environment variables.

You can use variables with expressions to conditionally assign values and further customize pipelines.

Use this YAML task template called SetBuildProperties.yml to add build properties in a pipeline:

SetBuildProperties.yml:

# Sets custom build properties. Use this to transfer info between pipelines.
parameters:

- name: buildId
  type: string

steps:
  
- task: PowerShell@2
  displayName: 'Set build properties'
  name: 'SetBuildProperties'
  inputs:
    targetType: 'inline'
    script: |

        # prepare the auth token
        $authHeader = "Bearer $(System.AccessToken)"

        $requestHeader = @{
            "Authorization" = $authHeader
            "Accept" = "application/json"
        }

        # array of operations
        # @{} is a hashtable in PowerShell. 
        # @() is an array (list) in PowerShell.
        $paramsObject = 
        @(
            @{
                "op" = "add"
                "path" = "/myProperty"
                "value" = "myValue"
            }
        )

        $params = ConvertTo-Json -InputObject $paramsObject

        Write-Host "JSON: " $params

        $contentType = "application/json-patch+json;charset=utf-8"        


        $apiurl = "https://dev.azure.com/MyOrg/MyProject/_apis/build/builds/${{ parameters.buildId }}/properties?api-version=6.0"
        
        Write-Host "Invoking REST API url: " $apiurl

        try 
        {
            $properties = Invoke-RestMethod -Uri $apiurl -Method PATCH -Headers $requestHeader -Body $params -ContentType $contentType

        }
        catch
        {
            # Writes an error to build summary and to log in red text
            Write-Host "##vso[task.LogIssue type=error;]Error occurred: " $_

            exit 1
        }

If you need to read the properties in the same or a different pipeline you can use this template GetBuildProperties.yml:

# Gets custom build properties. Use this to transfer info between pipelines.
parameters:

- name: buildId
  type: string

steps:
  
- task: PowerShell@2
  displayName: 'Get build properties'
  name: 'GetBuildProperties'
  inputs:
    targetType: 'inline'
    script: |

        # prepare the auth token
        $authHeader = "Bearer $(System.AccessToken)"

        $requestHeader = @{
            "Authorization" = $authHeader
            "Accept" = "application/json"
        }

        $apiurl = "https://dev.azure.com/MyOrg/MyCompany/_apis/build/builds/${{ parameters.buildId }}/properties?api-version=6.0"
        
        Write-Host "Invoking REST API url: " $apiurl

        try 
        {
            $properties = Invoke-RestMethod -Uri $apiurl -Method Get -Headers $requestHeader

            $propertiesJSON = ConvertTo-Json -InputObject $properties
            Write-Host 'properties: ' $propertiesJSON
        }
        catch
        {
            # Writes an error to build summary and to log in red text
            Write-Host "##vso[task.LogIssue type=error;]Error occurred: " $_

            exit 1
        }